%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Created on 2008-03-22 by ZHENG Zhong
% Last changed at $Date: 2008-04-28 10:15:35 +0000 (Mon, 28 Apr 2008) $ by $Author: heavyzheng $, $Revision: 39 $
% $HeadURL: http://buggarden.googlecode.com/svn/quant/study_notes/options_on_stock_indices_currencies_futures.tex $
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{Options on Stock Indices, Currencies, and Futures}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Options on Stocks Paying a Known Dividend Yield}

Consider the difference between a stock that pays a dividend yield at a rate $q$ per annum and an
otherwise identical stock that pays no dividends. Both stocks should provide the same overall return
(dividends plus capital gain). The payment of a dividend causes the stock price to drop by the
amount of the dividend. The payment of a dividend yield at rate $q$ therefore causes the growth rate
in the stock price to be less than it would otherwise be by an amount $q$. If, with a dividend yield
of $q$, the stock price grows from $S_0$ at time zero to $S_T$ at time $T$, then in the absence of
dividends it would grow from $S_0$ at time zero to $S_T e^{qT}$ at time $T$. Alternatively, in the
absence of dividends it would grow from $S_0 e^{-qT}$ at time zero to $S_T$ at time $T$.

This argument shows that we get the same probability distribution for the stock price at time $T$ in
each of the following two cases:

\begin{my_itemize}
  \item The stock starts at price $S_0$ and pays a dividend yield at rate $q$.
  \item The stock starts at price $S_0 e^{-qT}$ and pays no dividend yield.
\end{my_itemize}

This leads to a simple rule: when valuing a European option lasting for time $T$ on a stock paying a
known dividend yield at rate $q$, we reduce the current stock price from $S_0$ to $S_0 e^{-qT}$ and
then value the option as though the stock pays no dividends.

By replacing $S_0$ by $S_0 e^{-qT}$ in the Black-Scholes formulas, we obtain the price $c$ of a
European call and the price $p$ of a European put on a stock providing a dividend yield at rate $q$
as:

\begin{equation}
  c = S_0 e^{-qT} N(d_1) - K e^{-rT} N(d_2)
  \label{20080322_bs_call_dividend_rate}
\end{equation}

\begin{equation}
  p = K e^{-rT} N(-d_2) - S_0 e^{-qT} N(-d_1) 
  \label{20080322_bs_put_dividend_rate}
\end{equation}

Since:

\[ ln \Big( \frac{S_0 e^{-qT}}{K} \Big) = ln \frac{S_0}{K} - qT \]

it follows that $d_1$ and $d_2$ are given by:

\[ d_1 = \frac{ ln(S_0/K) + (r - q + \sigma^2/2) T }{ \sigma \sqrt{T} } \]
\[ d_2 = \frac{ ln(S_0/K) + (r - q - \sigma^2/2) T }{ \sigma \sqrt{T} } = d_1 - \sigma \sqrt{T} \]

Note that the word \emph{dividend} should be defined as the reduction of the stock price on the
ex-dividend date arising from any dividends declared. If the dividend yield is not constant during
the life of the option, equation \eqref{20080322_bs_call_dividend_rate} and
\eqref{20080322_bs_put_dividend_rate} are still true, with $q$ equal to the average annualized
dividend yield during the life of the option.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Options on Stock Indices}

In valuing index options, we may assume that the index could be treated as a security paying a known
dividend yield. This means that equations \eqref{20080322_bs_call_dividend_rate} and
\eqref{20080322_bs_call_dividend_rate} can be used to value European options on an index. $S_0$ is
equal to the value of the index, $\sigma$ is equal to the volatility of the index, and $q$ is equal
to the average annualized dividend yield on the index during the life of the option. The calculation
of $q$ should include only dividends whose ex-dividend date occurs during the life of the option.

If the absolute amount of the dividend that will be paid on the stocks underlying the index (rather
than the dividend yield) is assumed to be known, the basic Black-Scholes formula can be used with
the initial stock price being reduced by the present value of the dividends.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Currency Options}

\textcolor{red}{todo}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Futures Options}

\textcolor{red}{todo}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Black's Model for Valuing Futures Options}

\textcolor{red}{todo}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%











